In the Catan development card pile, there are 25 cards, 14 of them are type A, 5 of them are type B, 5 of them are type C, 2 in each type D, E, F. After a certain point in time, cards are drawn without replacement, so there are only $n \leq 25$ cards left in the pile. What is the probability of selecting a card of each type then, in terms of $n$?
I have tried using the sum of products between hypergeometric distribution with the probability drawing the exact card type, with the result as follow: $$\sum_{x=1}^a\frac{\binom{x}{a}\binom{20}{25-n-a}}{\binom{25}{25-n}}\frac{a-x}{n}$$ where $a$ is the number of card in each types (Ex $a=5$ for type B). My reasoning process is to multiply the probability that a certain amount $x$ of cards in each category being drawn with the probability drawing one of that type from the $n$ cards left, but I'm not sure if I made any mistake along the way. Also, I'm not sure if there are any other types of distribution available to generalized problems like this.
If you haven't looked at the cards that were already drawn, the chances on card $n$ are the same as on card $1$, so $\frac {14}{25}$ for $A$ and so on. If you have looked at some of the cards that have been drawn, you should work with the population of all the cards you don't know. So if you have drawn $3\ A$s and a $B$ while others have drawn $10$ cards you have not seen, the chance of an $A$ is $\frac {11}{21}$